7"tlin- Walled Structures 7 (1989) 213-238

Inelastic Buckling Analyses of Beams, Columns and Plates Using the Spline Finite Strip Method

S. C. W. Lau & G. J. Hancock* School of Civil and Mining Engineering. University of Sydney. New South Wales 21X)6.

Australia

(Received 15 February 1988: accepted 27 June 1988)

A BSTRA CT

A method of inelastic buckling analysis of thin-walled strtwtttral members and phttes is described. The method is based on the spline [inite strip method of structural analysis. The analysis takes into account the non-linear material stress-strain properties, strain hardening and residual stresses. The plastic theories used in the study are the flow theory of phtsticitv attd the deforma- tion theory of plasticity. The method of inelastic bttcklhzg analysis is appl&d to a variety of instability problems inchtding plates, cold-formed columns, hot-rolled cohtmns and welded tee section beams. The btwklhtg modes" and loads compttted are compared with theoretical vahtes and test resttlts.

N O T A T I O N

a Length of plate b Width of plate E, Es, Young's modulus and

respectively Fv Yield stress G Shear modulus l/r Column slenderness ratio Mc Critical buckling moment ME Elastic buckling moment Mp Plastic moment

strain hardening modulus,

*To whom correspondence should be addressed.

213 Thin-Walled Structures 0263-8231/89/$03.50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

214 S . C . W . Lau. G. J. Hancock

M~,AI,.,M~,

t

w

8M~, 8M,,, 8M~,. 8Xl, ,SX,_, 8X3

Ex, ey, "Y.~.v

Ey, Est

P

o"

O'cr

O" N

O'~, O" v . Txy

O'y

X~, X-', X3

Bending moments per unit width of plate in x. v directions respectively and twisting moment in x and v plane resulting from flexura[ buckling deformations Thickness of plate Deflection of plate

An infinitesimal change in Mx, M,, M~,, respectively An infinitesimal change in Xt, X2, X3, respectively Strain Strains in x, 3' directions respectively and shear strain Strains at yield and the onset of strain hardening, respectively Poisson's ratio Stress Critical buckling stress Maximum stress in triangular compressive edge load Stresses in x, y directions respectively and shear stress Yield stress Curvatures of plates in x, y directions respectively and twist in x and y plane

Subscripts e Elastic F Flexural M Membrane p Plastic x, v Rectangular Cartesian coordinates

1 INTRODUCTION

In this paper, a method of inelastic buckling analysis of thin-walled sections is described which takes into account the residual stress distributions and the non-linear stress-strain properties of the material from which the section is made. The analysis is based on the spline finite strip method ~ which has been shown to be both accurate and efficient for the elastic buckling analysis of thin-walled structural members and plates. 2 The spline finite strip method is more efficient than the finite element method 3 for analysing thin-walled sections. It retains some of the features of the semi-analytical finite strip metho& but allows boundary conditions other than simply supported ends.

The tangent modulus concept 5 has generally been accepted for inelastic buckling of columns. However, inelastic theories for plate buckling can be

hlelastic buckling of beams, coh~mns and plates 215

divided into two groups, namely, the deformation theory, of plasticity and the flow theory of plasticity. The flow theory 6 has been well accepted as a general and valid theory of plasticity while the deformation theory. 7'8 has limited applications. The flow theory of plasticity relates the increments of plastic strain to the stress so that the plastic strains depend on the loading history. The deformation theory of plasticity relates the total plastic strains to the current state of stress so that the plastic strains are independent of the loading history. The deformation theory of plasticity cannot be used for general applications because the plastic strains generally depend on the loading paths. 9 However. buckling loads obtained from the deformation theory for the inelastic buckling of plates were in better agreement with test results.

Both the flow and deformation theories of plasticity are summarised in this paper. The stress-strain relations derived from the two theories are used separately in the spline finite strip method of buckling analysis. The method is applied to a variety of instability problems including plates, columns and beams. The buckling loads and modes computed are compared with theoretical values and test results in order to check the accuracy and effi- ciency of the method for inelastic buckling analvsis of structural members and plates including sections with fixed ends.

2 INELASTIC BUCKLING BEHAVIOUR

2.1 General

Thin-walled steel structural members are generally manufactured by the process of hot-rolling or cold-forming. In cold-formed sections, the mechanical properties of the material are changed from the original state as a result of the cold work involved in the manufacturing process. A cold- formed steel section may have a gradual yielding stress-strain curve for the material forming the section. The proportional limit of the material depends not only on the amount of cold work but also on the amount of Bauschinger strain involved in the forming process.t° If the buckling stress of a section is higher than the proportional limit of the material making up the section, inelastic buckling occurs such that the rounded region of the stress-strain curve has to be taken into account in the analysis. In hot-rolled sections, residual stresses exist over the cross-sections because of the different cooling rates during the manufacturing process. The stress-strain curve of the material in a hot-rolled steel section usually remains linear until yielding occurs. Sections with intermediate slenderness buckle inelasticallv because of premature yielding of the cross-sections as a result of the residual stresses.

216 S. C. W. Lau, G. J. Hancock

2.2 Cold-formed sections

Cold work, either by the process of cold-rolling or brake-pressing, usuallv increases the yield and ultimate strengths and reduces the ductility of the material, especially at corners and bends of the sections. The stress-strain curve shown in Fig. la is a gradual yielding type because the various fibres in the material are cold-worked by different amounts and consequently yield at different stresses." The non-linear region of the stress-strain curve is also affected by the previous stress history of the material. As the steel strips are usually stored as coils for most light gauge members, flexural prestrains will be induced when the steel strips are flattened in the forming process. The tensile and compressive proportional limits may be different as a result of the Bauschinger effect. Therefore, the non-linear region of the stress-strain curve is important for sections with intermediate slenderness which buckle at a stress above the proportional limit.

2.3 Hot-rolled sections

For a hot-rolled I-section, the regions of residual compressive stress are near to the flange tips and the middle of the web, and the regions of residual tensile stress are near to the flange-web junctions. An idealised stress-strain curve for a hot-rolled steel is shown in Fig. lb. For an intermediate length compression member, the combined effects of the applied load and com- pressive residual stresses may cause premature yielding of a portion of the cross-section. With increasing applied load, the column will buckle inelastically at a load lower than the elastic buckling load as a result of the loss of stiffness due to the spread of yielding in the cross-section. 12

l /--- E

s r,s,l i _ / ~Proportionat Limit

Fy

Stress I

I I I ('r Strain ¢~,t

Strain (a) (b)

Fig. 1. (a) Stress-strain curve for cold-formed steel. (b) Idealised stress-strain curve for hot-rolled steel.

Inehtstic buckfing of beams, coh,mns and plates 217

3 STRESS-STRAIN RELATIONS

3.1 General

In the elastic range, the membrane stress-strain relations are given bv

O" x

O" v

r~y

E vE | - - V 2 l - - V 2

vE E I - - V 2 l - - V 2

0 0 E

2(I + v)

~x

~v

I (la)

which can be represented in matrix notation by

{or} = [OM]e {¢} ( lb)

The bending moments and twist ing moment in a plate as a result of buckl ing are related to its curvatures and twist bv

{M} = [D~]~ {X} (2a)

[Or]¢ = ~ [DM]¢ (2b)

where

{M} T = {M, MyM,y} (2c)

02 w 02 } w 32 w {X} r = {XtXzX3} = ax z a---7 2 axav (2d)

and the matrices [DM], and [DF~k are the elastic property matrices describing the membrane and flexural displacements respectively.

In the inelastic range, the incremental stress-strain relations for the case of uniform longitudinal compression in the x direction as shown in Fig. 2 can be expressed by

iix o1 :l{:t dory ~ = t Dy

dr~y j o D~y dy~y j

(3a)

218 S. C. U,'. Lau. G. J. H, mcock

which can be represented in matrix notation by

{do-} = [DM]p {de} (3b)

Similar to eqns (2a)-(2d), and assuming continuous loading across the plate thickness when buckling occurs, the change in bending moments and twisting moment in a plate can be related to its change in curvatures and twist by

{SM} = [Dv]p {6X} (4a)

? [Dv]p = [~[DM]p (4b)

where

{SM} T= { ~Mx ~My ~4xy } (4c)

{Sx} r = {Sx, Sx_,8X3} (4d)

and the matrices [DM] p and [DF] p are the plastic property matrices describing the membrane and flexural displacements respectively.

3 . 2 F l o w t h e o r y o f p l a s t i c i t y

According to the flow theory of plasticity as formulated by Handelman and Prager, 6 the material property moduli in the inelastic range. De. D,,. D~, and Dxv, are given by

K + 3 Dx - - - (5a)

fl

4K D y - ll (5b)

2(K - 1 + 2v) Dl = (5c)

11

E Dxy = G - 2 ( 1 1 v ---------~+ (5d)

K(5 - 4 v ) - ( 1 - 2v): fl = (5e)

E

E K - ( 5 f )

Inelastic buckling of beams, colurmzs and plates 219

q - -

~I o 4

I

i

a

J

i 0

Fig. 2. Plate subjected to uniform compression.

where E, is the tangent modulus determined from the stress-strain curve of the material, and v is the elastic value of Poisson's ratio. It should be noted that if r = l is substituted into eqns (5), then [DM]~ = [DM]p and [D~]~ = [D~]p.

3.3 Deformation theory of plasticity

Inelastic plate buckling problems have been investigated bv Bijlaard, 7 Stoweli 8 and Ilyushin ~3 with the stress-strain relations based on the deformat ion theory, of plasticity. Bijlaard and Stowell assumed continuous loading across the thickness of plate when buckling occurred, whereas llyushin assumed the concept of strain reversal on the convex side of the plate. Test results agreed better with the analyses of Bijlaard and Stowell, therefore their analyses are summarised in this paper.

3.3.1 Bijlaard's analysis

The material property moduli according to Bijlaard 7 are given bv

K + 3 + 3 e Dx - (6a)

fl

4K D y - fl (6b)

2(K- 1 + 2v) Di = (6c)

fl

E D~r - 2(1 + v) + 3e (6d)

._0 S. C. W. Lau, G. J. Hancock

9l = K ( 5 - 4 v + 3 e ) - ( l - 2v): (6e) E

E e = - - - 1 (6f) Es

where Es is the secant modulus determined from the stress--strain curve of the material, v is the elastic value of Poisson's ratio and K is as defined in eqn (5f). Equations (6) will be the same as eqns (5) if e = 0 is substituted. Also, if E = E, = E~ is substituted into eqns (6). then [DM]~ = [Dx,]p and [D ]o =

3.3.2 Stowell's analysis

The material property moduli according to Stowell ~ are given bv

K + 3 + 3 e D x - (7a)

4K (7b) Dy -

2K Dl - (7c)

fl

E (7d) Dxy - 3 + 3e

lq - K(3 + 3e) (7e) E

where K and e are as defined in eqns (5f) and (6f) respectively. If v = 0.5 is substituted into eqns (6), it will be the same as eqns (7). It should be noted that [DM]~ :~ [DM]p and [DF]~ ~: [DF]p if E = E, = E, is substituted into eqns (7).

4 SPLINE FINITE STRIP BUCKLING ANALYSIS

4.1 Formulation

The spline finite strip method has been applied by the authors for elastic buckling analysis of thin-walled sections subjected to compression, bending and shear. A detailed description of the method can be found in Lau and Hancock. 2 In the semi-analytical finite strip method developed by Cheung, ~

Inelastic buckling of beams, columns and phltes 221

the displacement functions of a strip are expressed as the product of trans- verse interpolation polynomials and longitudinal trigonometric series. The spline finite strip method is similar to the semi-analvtical method except that the longitudinal trigonometric series is replaced bv a linear combination of local B3-splines. ~ Consequently, due to the Iocalised nature of the B3-spline function, the method is not restricted to structures with both ends simply supported as in the semi-analytical method. Different boundary conditions can be specified by modifying the local B:-splines.

In the spline finite strip method of buckling analysis, both membrane and flexural displacements have been included in the formulations. In the elastic range, the element stiffness matrices describing the membrane and flexural displacements are given bv eqns (8a) and (,qb) respectively.

[Kv]¢ = f . [BF]T[Dv]~[BF]dV (8a)

[Kr~t]~ = f, [B,~,]T[DM],[BM]dV (Sb)

where IBM] and [BF] are the strain matrices describing the membrane and tlexural displacements respectively. Using the principle of minimum potential energy, the elastic stability problem is characterised by the follow- ing eigenvalue equation.

( [ K ] 0 - X [ G ] ) ' ' = ' ' (9)

where

[K]¢ = global elastic stiffness matrix

[G] = global stabilitv matrix

{81 = displacement vector

h = buckling load factor

In the plastic range, the elastic property matrices in eqns (8) have to be replaced by the corresponding plastic property matrices. Therefore. the plastic element stiffness matrices are given bv

' [K r ]p = f , [BF]r[Dv]p[Bv]dV ( lOa)

[ K M ] . = (10b)

222 S.C. t,$'. La,, G. J. Hancock

The plastic stability problem is then represented by the following matrix equation

([K]~ - x [ G ] ) { 8 1 = {01 (1l)

where [ K]p is the global plastic stiffness matrix. The matrices [Kv]p and [KM]p are not constant because the plastic

property matrices depend on the stress level. Depending on the theory of plasticity employed, the plastic property matrices can be determined from either eqns (5), (6) or (7). The material property moduli as formulated by Bijlaard 7 (eqn (6)) have been used in the spline finite strip buckling analysis computer program since they can be easily modified to eqns (5) and (7), and they are still valid in the elastic range.

The method of solution of eqn (I 1) and the representation of the stress- strain curve of the material are discussed in the following two subsections.

4.2 Solution of eigenvalue problem

For an elastic buckling analysis.-" the eigenvalue problem in eqn (9) has been solved by a method which is based on the Sturm sequence property with the eigenvalues extracted by a bisection strategy.~5 With a trial value of h. the method involves reduction of [K]. - h[G] to an upper triangular form bv a simple Gaussian elimination without row interchanges. The number of eigenvalues less than the trial value of k is equal to the number of negative diagonal elements in the triangulated form of [K]~ - h[G]. Knowing this information, a new trial value of h is chosen and the triangulation process is repeated until the required eigenvalue corresponding to a particular mode number is isolated. The choice of the trial values of h is based on a bisection method, and the convergence criterion is satisfied when two successive trial values of h are smaller than a specified value.

For inelastic buckling analysis, the critical state of stress is when ,k equals one. One of the methods which Pifko and Isakson t6 used to solve this problem was by a process of trial and error. In this method the matrices [ Kip and [G] are computed at a trial loading level, and the corresponding lowest eigenvalue is then determined. If the eigenvalue so determined equals one, then the trial loading is the critical one. If it is greater or less than one, the current trial loading level is not critical and the next trial loading has to be increased, or decreased. Obviously, this method is rather inefficient because it involves repeatedly solving the eigenvalue problem at every trial loading level.

Gupta t7 presented an efficient method which was also based on the Sturm sequence property employing a bisection strategy. This study also uses the method presented by Gupta but with some modifications.

Inelastic buckling of beam.s', columns antt plates 223

The me thod involves, firstly, substituting h = I and a trial load into eqn (11). The usual procedure of reduction of [ K ] p - X [ G ] to an upper t r iangular form is performed. The number of eigenvalues less than one, say NR, is de te rmined by counting the number of negative diagonal elements. If NR equals zero, the current stress state is less than the critical state, and the next trial load will be increased from the current stress state. If NR is equal to or greater than one, then the current stress state exceeds the critical state, and the next trial load will be decreased instead. With a new trial load, the process of tr iangulation is repeated until convergence takes place. The convergence criterion is satisfied when the ratio of two successive trial loads is less than, say, 0.01%. The choice of the trial loads is based on a bisection strategy, and the first trial load is chosen as

Py P t - 9 (12)

where Pv is the squash load. Consequent ly , with some minor modifications, the eigenvalue routine

used in Lau and Hancock-" can still be applied to inelastic buckling problems.

4.3 Stress-s t rain curve

In o rde r to de te rmine the values of the tangent modulus and secant modulus at different stress levels, non-linear material stress-strain curves have been represen ted in the forms of formulae by different investigators and are summar i sed as follows.

4.3.1 Chajes. Fang and Winter Accord ing to Chajes et al.tS stress-strain curves can be represented by For o- < 0,667o-y:

E = E~ = E, (13a)

For o- -> 0.667o-v:

o'v In ( - - ~ _ ~ ) 0.513o'v E - 4 .5 -~ + ~ (13b)

dE (13c)

The value of 4-5, which was based on the statistical mean from the test results, was chosen for cold-formed light gauge steel sections. The shape of the stress-strain curve de te rmined using eqn (13) is demonst ra ted in Fig. 3a where E = 2 x 1@ MPa and o'~ = 310 MPa have been used.

__4 S. C. [L. Latt, G. J. Hancock

I / F io | / E :2xlOsMPa / / o~:310.Pa

"-. 0 5 F / "~ 0.5 ~" I / Equahons 13a-13c

I / ! I i I I I I

0 1 2 3 z+ 5 6XI0 3

Strain

(a)

1.0

"-. 0.5

!

SMPa MPa

Equations 1(,a- IL, b

R a m b e r g -

/ / , " (c=0.95)

E=ZxIOSMPa / Oy=310MPa

/ 0o7=0 8Boy oo.ss:O 770y

/ Equations 15b-15d

I I I I i 1 2 3 t, 5 6x10 "3

S t r a i n

(b)

I I I I I I 1 2 3 t, 5 6x10 -3

Strain

(c)

Fig. 3. Stress-strain curves according to (a) Chajes. Fang and Winter tS, (b) Plank 19 and (c) the Ramberg-Osgood formula. ~

4.3.2 Plank According to Plank, ~9 stress-strain curves can be represented by

o- (1 - c / t ) ( 14a ) E - -

E ( l - ~t)

do- _ E ( I - p.)z ( 14b ) d~ 1 - 2c/z + c/z:

w h e r e / x = cr /o 'v and c is a constant. A value o f c = 0 . 9 9 7 has been sug- gested for structural steel. Figure 3b shows the effects of the c values on the shapes of the stress-strain curves determined using eqn (14). Again, E = 2 x llY MPa and o'v = 310 MPa have been used in Fig. 3b.

4.3.3 Ramberg-Osgood The general form of the Ramberg--Osgood formula is

E = ~ - + k (15a )

bwhtstic bucklb~g of beams, coh+mns and ph+tes 225

where k and n are constants. Ramberg and Osgood 2° have derived expres- sions for determining k and n, hence, eqn (15a) can be re-written as

o. 3o. ( o. / " - t E ---- -~ - q"

" ~ \ o.o.7 ] (15b)

do- E d~

(15c)

where

n = 1 4 1og(17/7) (15d) Iog(o.o.7/o'0.85 )

and the stresses o'0.7 and 0"o.+5 are determined from the stress-strain curve of the material. Thev are the stresses corresponding to E+ = 0.7E and E+ = 0-85E respectively. The shape of the stress-strain curve determined using the Ramberg-Osgood formula is demonstrated in Fig. 3c where the formula has been used to represent the stress-strain curve in Fig. 3b with c = 0.95. The values of o-o.7 and O'o.85 are shown in Fig. 3c. The curve determined using the Ramberg-Osgood formula matches Plank's curve when 0./0.v is less than 0.9 approximately but keeps increasing for larger values of 0./0.y.

5 NUMERICAL INVESTIGATIONS

5.1 General

The validity and accuracy of the spline finite strip method of inelastic buckling analysis described in the previous section were investigated by comparing the buckling loads and modes computed with the theoretical values and test results. The method was applied to a variety of instability problems including plates, columns and beams. The buckling loads obtained using the plasticity model based on the deformation theory are compared, in some cases, to those based on the flow theory. Since the inelastic stiffness matrices of the spline finite strip method have been formulated with the assumption that the stress distribution is uniform within a strip, therefore in the following analyses, the linear distributions of residual stress and bending stress have been approximated in the form of finite jumps over the plate width.

226 S. C. W. Lau, G. J. Hancock

5 . 2 P l a t e s

T h e inelast ic buck l ing of s imply s u p p o r t e d and c l a m p e d pla tes has been

s tud i ed by P i fko and I sakson ~6 using the finite e l e m e n t m e t h o d with ma te r i a l

p las t ic i ty b a s e d on the d e f o r m a t i o n theory . T h e e l e m e n t type used was a

r e c t a n g u l a r e l e m e n t o f 16 degrees o f f r e e d o m with compa t ib i l i ty for bo th

s lope and de f l ec t ion a long in t e r - e l emen t boundar i e s . T h e buckl ing s t resses c o m p u t e d us ing the spl ine finite str ip m e t h o d are c o m p a r e d with those

TABLE 1 Critical Stresses of Simply Supported Square Plates under Uniform Compression

(a = h = 20in.}

Finite element Spline finite Thickness anah'sis strip amdvsis

(in.) (psi) (psi)

11.778 67 65 01)2 65 l~12 (I.858 011 75 0113 75 1~12 (7.964 49 85 (~13 85 0011 I" 12(1 19 95 0112 95 1~)0 1-366 78 1115 0112 1115 (R)O 1-767 52 1 15 01)0 I 15 0~10 2-390 53 125 002 125 0011

I psi = 6.895 x 10 -3 MPa. 1 in. = 2 5 . 4 m m .

T A B L E 2

Critical Stresses of Simply Supported Rectangular Plates under Uniform Compression (a = 30 in.b = 20 in.}

Finite element Spline finite Thickness analvsis strip amdysis

(in.) (psi) (psi)

0-750 88 65 005 65 014 0-835 18 75 01}4 75 01)7 0-954 29 85 004 85 004 1.127 10 95 002_ 95 009 1.390 64 105 001 105 006 1.808 84 115 001 115 004 2.453 21 125 001 125 011

1 psi = 6.895 x 10 -3 MPa. I in. = 25-4 mm.

Inehtstic buckling o f beams, columns and phues 227

T A B I . E 3

Cri t ica l S t resses o f C l a m p e d Squa re Pla tes u n d e r U n i f o r m C o m p r e s s i o n (a = h = 211 in.)

Finite element Spfine finite Ditlerence Thickness atlalvsi.s strip anah'sis as a

tin. ) (psi t ( psi t percentage

1/-5 66 414 66 419 II.ll 11-6 SI 712 ,~1 71S II-l} 11.7 91 234 91 0Oll 11.2 (I-S 97 549 97 IN) Ib4

I psi = 6.895 x 11) -3 MPa .

I in. = 2 5 . 4 m m .

obtained by Pifko and Isakson in order to check the efficiency and accuracy of the method for inelastic buckling analysis.

Based on Ramberg-Osgood representation for the stress-strain curve of the material, Pifko and Isakson computed the inelastic buckling stresses of simply supported square plates, simply supported rectangular plates and clamped square plates subjected to uniform compression. They are compared in Tables 1, 2 and 3 with those obtained using the spline finite strip method. The material properties are as follows.

E = 10 7 psi (16a)

, = 0-5 (16b)

O'o.7 = 105ps i (16c)

o'0.85 = 0"906O'o.7 ( 1 6 d )

In the finite element analyses, a 4 x 4 mesh was used for the simply supported square plates, a 6 x 4 mesh for the rectangular plates and a 6 x 6 mesh for the clamped plates. Because of symmetry, only one-quarter of the plates was analysed. In the spline finite strip analyses, for one-quarter of the plates, 3 strips x 3 spline sections were used for the simply supported square plates, 3 strips x 4 spline sections for the rectangular plates and 5 strips x 5 spline sections for the clamped plates. The finite element method was also applied to simply supported square plates subjected to triangular compres- sive edge load where a 8 x 4 mesh was used for half of the plates. In the corresponding spline finite strip analyses, the plates were subdMded into 8 strips x 4 spline sections for half of the plates, and the idealisation of the

228 S. C. ~V. Lau. G. J. Hancock

TABLE 4 Critical Stresses (Cry) of Simply Supported Square Plates under Triangular G~mprcssi~c

Edge Loads (a = /9 = 20in.)

Finite eh'ment Sl,line linite Thickness analvsi,s strip amdysis

(in.) q~si~ cp~i)

0-5 53 483 53 522 0.6 76 646 7f~ 7()1 0.7 I(X) 155 ll)~) 22S 0.8 117 923 117 9Sl 1.0 138 749 138 ,~()1

1 psi = 6.895 x 10 -3 MPa. I in . = 2 5 . 4 m m .

stress distribution was the same as that used by Pifko and Isakson. The buckling stresses computed are summarised in Table 4.

It can be seen from Tables 1 to 4 that the buckling stresses obtained using the spline finite strip method agree closely with those obtained using the finite e lement method with a maximum difference of 0-4%.

5.3 Columns

5.3.1 Flexural buck l ing

The spline finite strip method was applied to a simply supported lipped channel section column subjected to uniform compression so as to compare the computed inelastic flexural buckling stresses with the theoretical values. The dimensions of the channel section and the strip subdivision of the section for the analyses are shown in Fig. 4. Because of symmetry, the column was subdivided into 4 spline sections longitudinally for half of the length of the column. The stress-strain curve of the material used for the analyses was based on eqns (14) with E = 2 x l0 s MPa, cry = 310 MPa and c = 0.95. The value of Poisson's ratio has been taken as 0.3. The computed inelastic flexural buckling stresses with the plasticity model based on the deformat ion theory are compared in Table 5 with those based on the flow theory for different values of column length. The theoretical buckling stresses obtained using the following tangent modulus equation are also included in Table 5.

(17)

Inehtstic buckling of beams, columns and plates __9

F

I 9oi t=l S Dimensions In mm

50 _ I

Fig. 4. Dimens ions of lipped channel section column and subdivision of strips for inelastic flexural buckling analysis.

TABLE Inelastic Flexural Buckling Stresses of Lipped Channel Section Column

Spline finite strip analysis

Deformation Flow Tangent modulus Length theory theory hucklin~ ~'tress (mini (MPa) (~lPa) (MPa)

411t) 293.36 294-57 292 "35 71111 278.66 279-15 279.93

I II11) 254-69 254"95 254.14 I 500 2211.65 220.84 220. IS 2 ()1~) 163.78 163.86 163-19

It can be seen from Table 5 that the computed values agree with the tangent modulus values. It is also interesting to note that the computed values based on the flow theory agree with those based on the deformation theory since the shear modulus does not play a significant role in flexural buckling.

5.3.2 Cold-formed sections Chajes et al. 18 tested a variety of cold-formed columns with both ends fixed which failed by elastic or inelastic flexural-torsional buckling. The tested sections included plain angles, lipped angles, channels and hats whose dimensions and the measured yield stresses are also summarised in Lau and Hancock. -'t Chajes et al. calculated the inelastic flexural-torsional buckling loads by

230 S. C. W. Lau. G. J. Hancock

TABLE 6 Elastic Flexural-Torsional Buckling Tests by Chajes et al. Compared with

Spline Finite Strip Values

Specimen

Experimental Chajes et al. Splme strip critical theoretical buckling stress buckling stress Ratio stress

¢rE (ksi) rrr rksi) ~rLhrr ,rs tksi) Ratio CTEiCTS

Plain angle A-E I 24.4 23.8 1-03 25.11 0.98

Lipped angle LA-E 1 17.4 17.7 0-98 17.S 0.9,'4

Channel CH-E 1 16.5 17.6 0-94 17.(1 0.97

Hat H-E 1 12-3 13.4 I).92 12.0 1 -I)3

lksi = 6.895 MPa.

t ' n I I J I

(al AngLe (t'/ L~ppe~ ,~ngte

t I I I 0

0 | I

1¢) Channe[ (d) Hat

Fig. 5. Finite strip subdivision of sections tested by Chajes, Fang and Winter. is

Inelastic buckling of beams, coh,mns and plates 231

TABLE 7 Inelastic Flexural-Torsional Buckling Tests by Chajes et al. Compared with

Spline Finite Strip Values

Specimen

Experimental Chajes et al. critical theoretical stress buckling stress Ratio

(re (ksD (rr (ksD (re/(rr

Splhte strip buckling

stress (deformation

theory) (rs {ksi)

Ratio (rE/(rs

Plain angle A-I I 38.3 36.7 1.04 36-7 1.()4 A-I2 38.3 37.5 1.1)2 40-3 0.95 A-I3 33.2 29.9 1-11 34-7 11.96 A-I4 33.2 30.4 1-09 34. l 0.97 A-15 32.2 28.9 I- I 1 32-2 I .(~) A-J6 22.5 24.5 0.92 26.9 I).84

Lipped angle LA-II 26.6 37.2 I)-72 39-7 I)-67 LA-I2 34.6 38-8 0.89 36-4 I).95 LA-[3 31)-3 32-7 I).93 34.7 0.87 LA-14 30-2 31.3 0-96 31.8 0.95 LA-I5 26-5 25. I 1.1)6 26-5 1.00 LA-I6 23.0 24.7 1.05 28.2 I).82 LA-I7 25.0 26.4 I).95 27.5 (1.91 LA-I8 27-2 28.7 0.95 29.7 t).92 LA- I9 26.3 26.2 14)0 27.7 I).95

Channel CH-I 1 38.8 40.8 I).95 39.5 0.98 CH-I2 29.5 28.5 1-1)4 29.0 1.1)2 CH-I3 27.9 27.9 14)0 28.2 11.99 CH-I4 25.0 25.9 0.97 24.9 14]0 CH-I5 25.2 26.3 0.96 25-5 0.99 CH-I6 27.3 26-9 1.01 26-6 1.03 CH-I7 27.0 26.9 1.00 26.6 1-02_ CH-I8 26-4 26.9 0.98 26.6 I).99 CH-I9 26.7 27.1 I)-99 26.6 1.00

Hat H-I I 38.2 39.6 0.96 40-6 (]-94 H-I2 39.2 39-7 0.99 4 I- I 0.95 H-I3 31-6 34-5 0.92 35-6 0.89 H-I4 23.8 28.0 0.85 28.6 0.83 H-I5 22.6 26-8 0-84 27.1 I).83 H-I6 23-3 24-7 0.94 25.6 0.91

I ksi = 6.895 MPa.

232 S. C. W. Lau, G. J. Hancock

where P/, and P , are the inelastic and elastic tlexural-torsional buckling loads respectively computed accounting for the fixed end boundary conditions. Equations (13) were used to represent the stress-strain curve of the material. The theoretical elastic and inelastic buckling stresses calcu- lated by Chajes et al. are given in Tables 6 and 7 respectively and they are in close agreement with their test results.

The elastic and inelastic buckling stresses of these sections computed using the spline finite strip method with material plasticity based on the deformation theory are also included in Tables 6 and 7 respectively. The value of Poisson's ratio has been taken as 0-3 in the analyses. The strip subdivisions of the sections for the analyses are shown in Fig. 5, and because of symmetry, 4 spline sections were used for half of the length of the columns. The computed values are in close agreement with the test results for both the elastic and inelastic cases. The inelastic buckling stresses have also been computed using the plasticity model based on the flow theory. It was found that the values were close to those based on the deformation theory, with a difference of approximately !%, indicating that the shear modulus does not play a significant role in the computation of the buckling stress of these sections. It should be pointed out that the semi-analytical finite strip method cannot handle the fixed end conditions of these sections. The ability of the spline finite strip method to account for boundary conditions other than simply supported ends has been demonstrated.

5.3.3 Hot-rolled sections Kitipornchai and Lee-': performed a series of tests on simply supported angle and tee struts which failed by inelastic flexural buckling or inelastic flexural- torsional buckling. The residual stress distributions assumed in their analyses are shown in Fig. 6. The dimensions of their test sections are also given in Lau and Hancock. :~ The stress-strain curve of the material was assumed to be the same as that in Fig. lb. The properties of the material are as follows:

E = 2.14x10 -*MPa (19a)

~v = 289MPa (19b)

E~, = 0-0 (19c)

The test results and the theoretical buckling loads calculated by Kitipornchai and Lee are given in Table 8. The buckling loads of these sections computed using the spline finite strip method with material

Inelastic buckling of beams, columns and phttes _a3

TABLE 8 Inelastic Buckling Tests on Angles and Tee Struts by Kitipornchai and Lee Compared x~ith

Spline Finite Strip Values

Kitipornchai Spfne strip Experimental and Lee buckfing hind

fitilure theoretical ( de]brmathm load buckling load Ratio theoo') Ratio

Specimen PE (kN) Pr (kN) PE/PT Ps (kN) PE/Ps

Equal angle SAI 163 (F) 160 (F) I-(12 156 (F) 1-04 SA2 169 (F) 157 (F) 1.0g 156 (F) 1.0S SA3 t82 (F) 1911 (F) 0.96 195 (F) (I.93 SA4 167 (F) 182 (F) 0.92 169 (F) {I-99 SA5 161 (F) 17g (F) (1.91) 166 (F) 0.97 SA8 295 (F) 293 (F) 14)1 296 (F) I.(IO SA7 283 (F) 2911 (F) (I-98 2g~ (F) 0.9S

Unequal angle SAg 141 (FT) 128 (FT) 1-10 126 (FT) 1.11 SA9 135 (FT) 116 (FT) I. 16 117 (FT) 1" 15 SA 10 143 (FT) 140 (FT) 1.02 13(1 ( IT) I. 1(1 SA I 1 123 ( IT) 129 (FT) 0.95 1311 (b--'Y) 0-95 SA 12 277 (FT) 255 ( IT) 1-119 244 ( IT) I' 14 SA 13 1911 (Iq') 235 (FT) O-Sl 244 (FT) I).7,',I Tee

TI 315 (FT) 320 (FT) (I.98 339 (FT) (I.93 T2 320 (FT) 310 (FT) 1.03 339 ( IT) (}.94 T3 371 (F) 410 (F) 0.9() 399 (FT) 0.93 T4 385 (F) 399 (F) 0.97 399 ( IT) I)-96 T5 519 ( IT) 566 (FT) 0.92 543 (FT') (I.96 T6 513 (F'I') 556 ( IT) 0.92 543 ( IT) 0.94

(F) = flexural buckling. (FT) = flexural-torsional buckling.

plasticity based on the deformation theory are also compared in Table 8 with the theoretical values and test results of Kitipornchai and Lee. In the spline finite strip analyses, the Poisson's ratio has been taken as 0.3 and the values of E, and Es in the yielded regions were determined using eqns (20). Equation (20b) was derived by Ueda and Tal ly

E, = 0 .0 (20a)

Es _ O'y

__~4 S. C. W'. Lau. G. J. Hancock

/

(a) Assumed Residual Stresses

1 i 1 1 1 i i i i

I " i l l l i l l l l

(b) Strip Subdivision

Fig. 6. Angle and tee sections studied by Kitipornchai and Lee."-"

In eqn (20b), O'r is the residual stress and o-a is the applied stress. The strip subdivisions of the sections for the analyses are shown in Fig. 6, and because of symmetry, 4 spline sections were used for half of the length of the columns. It can be seen in Table 8 that the computed values are in close agreement with the theoretical values and test results of Kitipornchai and Lee. The buckling loads have also been computed using the plasticity model based on the flow theory. It was found that the buckling loads were identical to those based on the deformation theory.

5.4 Beams

Kitipornchai and Wong-Chung 24 investigated the inelastic buckling of welded monosymmetric I-beams subjected to uniform bending moment . The welding residual stress distribution assumed in their analysis for a tee section, which was based on the tendon force concept, is shown in Fig. 7. With the yield stress taken as 250 MPa and the stress--strain curve of the material assumed to be the same as that in Fig. lb but with Es, = 0, the

Inelastic buckling of beams, columns and plates 235

0 c I ' /

bf xi'f oc~w C dwxtw -'A

\

Fig. 7. D i m e n s i o n s o f tee sec t ion beam and residual s t resses a s sumed by Ki t iporncha i and W o n g - C h u n g . 24 bf = 146.1 m m , dw = 243.2 m m . t / = 8.64 ram, t,. = 6.1 ram.

Co = 20-0 m m , crtw = 50"82 M P a . cy,.~ = 44.8 MPa . crc = 95.08 M P a and tr>. = 250 MPa .

IE

10

081

06

0L,

02

A~ ~ ~ '

A 1 \~.....EL~st,c L. '\ BUckling

\ \ \

Kihpornchai and Wong-Chung • Sphne Finite Strip Analysls

I T ! 0S I0 IS

Modified Slenderness v~p/M E 20

Fig. 8. Comparison of inelastic buckling moments of tee section beam.

236 S. C. B,'. Lau, G. J. Hancock

theoretical critical moments calculated by Kitipornchai and Wong-Chung for the tee section in Fig. 7 have been plotted in Fig. 8. The critical moments of the tee section computed using the spline finite strip method with material plasticity based on the deformation theory, and u = 0.3, are also compared in Fig. 8 with the theoretical critical moments of Kitipornchai and Wong- Chung. In the analyses, the flange outstand and web plate were subdivided into 4 strips and 10 strips respectively, and because of symmetr), 4 spline sections were used longitudinally for half of the length of the beams. The computed critical moments are in the range of 3-10% lower than the values calculated by Kitipornchai and Wong-Chung. The critical moments have also been computed using the spline finite strip method but with material plasticity based on the flow theory. The values obtained were also identical to those based on the deformation theory.

6 CONCLUSIONS

Material plasticity has been incorporated in the spline finite strip method which has been previously developed for the elastic buckling analysis of thin-walled sections. The plastic theories investigated were the flow theory of plasticity and the deformation theory of plasticity. The analysis takes into account the non-linear nature of the stress-strain curves in cold-formed sections, strain hardening of the material and residual stress distributions in hot-rolled sections. The spline finite strip method, with the solution routine based on the modified form of Gupta's method, has proven to be both accurate and efficient for inelastic buckling analysis when the computed buckling loads and modes were compared with the theoretical values and the test results, including cold-formed columns, hot-rolled columns and welded tee section beams. Although the inelastic buckling loads of plates determined using the plasticity model based on the flow theory were much higher than those based on the deformation theory, it was found that the inelastic buckling loads obtained using the two theories were similar for sections which buckled in a flexural or flexural-torsional mode.

ACKNOWLEDGEMENTS

This paper forms part of a programme of research into the stability of steel structures being carried out in the School of Civil and Mining Engineering of the University of Sydney. The calculations were performed in the C. A. Hawkins Computing Laboratory at a terminal to a multi-user PRIME 9750 minicomputer. Funds to purchase the system were provided by the University and by the Civil Engineering Graduates' Association.

Inehtstic bucklin~ of beams, columns and phltes 237

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-~-~, _. 8 S. ('. ~. l.au. (;. J. Hapwock

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